The Lanczos method solves the eigenvalue problem
through partial tridiagonalizations of the matrix **C** using
Krylov bases. Unlike
direct factorization methods (e.g., QR factorization),
no intermediate dense submatrices (resulting from fill-in) are generated. Also,
information about **C**'s extremal eigenvalues tend to emerge long
before tridiagonalization is complete. Hence, the Lanczos algorithm is
particularly useful in situations where only a few of **C**'s largest or
smallest eigenvalues are desired.

An alternative method (referred to as CSI-MSVD) for
producing tridiagonal matrices whose eigenvalues approximate those
of the original matrix **C** was presented in
[10]. This method, based on
the extraction of modified moments from the Chebyshev semi-iterative
method [23], is an attractive alternative given its
scope for parallelism and reduced memory requirements.

Michael W. Berry (berry@cs.utk.edu)

Sun May 19 11:34:27 EDT 1996